Another case that yields (3) is that of a constant source distribution 



on a circle. This gives a velocity potential due to a source of strength 



2'n'ma at the center of the circle of radius a, and hence V'(s,n) '=. V'(s,0) = 



= 2Trm. But the local element about the point s contributes irm 



a 



to V' . Hence, if we consider a nonuniform distribution which is large in 

 the neighborhood of s, and negligible elsewhere, we would obtain V'(s,n) = 

 TTm, half of the Preston-Lighthill formula. This suggests that, without the 

 equipotential assumption, Equation (3) would give a good approximation only 

 for very thin two-dimensional forms. 



The distribution m(x) on the surface of the body y = f(x), which 

 satisfies the boundary condition 9(})/3n = v'(s,0) can be determined as the 

 solution of the Fredholm integral equation of the second kind, (see 

 Figure 2) 



Figure 2 



TTm(s) + (Jl m(t) -^ In r^ dt = V (s) (5) 



in which t also denotes arc length along the body contour. The indetermi- 



nacy of the kernel 7; — £n r at s = t can be removed by applying, the 

 dn st 

 s 



identity (at a smooth point of the contour) 



£,n r dt = (i) V- ^ dt = TT (6) 



St T 9t ^st 

 t 



St 



where the complex vector r e , extending from s to t, has given the 



pair of conjugate functions 



